On the periodic points of functions on a manifold
Author:
Chung-wu Ho
Journal:
Proc. Amer. Math. Soc. 130 (2002), 2625-2630
MSC (2000):
Primary 37C25; Secondary 54H25, 58C30
DOI:
https://doi.org/10.1090/S0002-9939-02-06361-X
Published electronically:
February 12, 2002
MathSciNet review:
1900870
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In a conference on fixed point theory, B. Halpern of Indiana University considered the problem of reducing the number of periodic points of a map by homotopy. He also asked whether the number of periodic points of a function could be increased by a homotopy. In this paper, we will show that for any map on a closed manifold, an arbitrarily small perturbation can always create infinitely many periodic points of arbitrarily high periods.
- 1. Louis Block, John Guckenheimer, Michał Misiurewicz, and Lai Sang Young, Periodic points and topological entropy of one-dimensional maps, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 18–34. MR 591173
- 2. B. Halpern, The minimal number of periodic points, Abstr. Amer. Math. Soc., 1 (1980), 775-G8, p. 269.
- 3. Chung-Wu Ho, On Block’s condition for simple periodic orbits of functions on an interval, Trans. Amer. Math. Soc. 281 (1984), no. 2, 827–832. MR 722777, https://doi.org/10.1090/S0002-9947-1984-0722777-3
- 4. Chung Wu Ho and Charles Morris, A graph-theoretic proof of Sharkovsky’s theorem on the periodic points of continuous functions, Pacific J. Math. 96 (1981), no. 2, 361–370. MR 637977
- 5. Morris W. Hirsch, Differential topology, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 33. MR 0448362
- 6. T. Y. Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985–992. MR 385028, https://doi.org/10.2307/2318254
- 7. James R. Munkres, Elementary differential topology, Lectures given at Massachusetts Institute of Technology, Fall, vol. 1961, Princeton University Press, Princeton, N.J., 1966. MR 0198479
- 8. Zbigniew Nitecki, Topological dynamics on the interval, Ergodic theory and dynamical systems, II (College Park, Md., 1979/1980), Progr. Math., vol. 21, Birkhäuser, Boston, Mass., 1982, pp. 1–73. MR 670074
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37C25, 54H25, 58C30
Retrieve articles in all journals with MSC (2000): 37C25, 54H25, 58C30
Additional Information
Chung-wu Ho
Affiliation:
Department of Mathematics, Southern Illinois University at Edwardsville, Edwardsville, Illinois 62026 – and – Department of Mathematics, Evergreen Valley College, San Jose, California 95135
Email:
cho@siue.edu
DOI:
https://doi.org/10.1090/S0002-9939-02-06361-X
Keywords:
Manifolds,
periodic points,
homotopy,
digraphs
Received by editor(s):
February 9, 1999
Received by editor(s) in revised form:
April 1, 2001
Published electronically:
February 12, 2002
Communicated by:
Alan Dow
Article copyright:
© Copyright 2002
American Mathematical Society
